Goal Recognition Design (GRD) problems involve identifying the best ways to modify the underlying environment that the agents operate in, typically by making a subset of feasible actions infeasible, in such a way that agents are forced to reveal their goals as early as possible. Thus far, existing work assumes that the outcomes of the actions of the agents are deterministic, which might be unrealistic in real-world problems. For example, wheel slippage in robots cause the outcomes of their movements to be stochastic. In this paper, we generalize the GRD problem to Stochastic GRD (S-GRD) problems, which handle stochastic action outcomes. We also generalize the worst-case distinctiveness (wcd) measure, which measures the goodness of a solution, to take stochasticity into account. Finally, we introduce Markov decision process (MDP) based algorithms to compute the wcd and minimize it by making up to k actions infeasible.